Christoffel Symbols

Every numerical relativity code that models a black-hole merger begins by computing Christoffel symbols. When LIGO and Virgo detect a gravitational wave, the template bank it is matched against comes from simulations that integrate Einstein's equations on a grid, and at each of billions of grid points the machine evaluates $\Gamma^{\lambda}{}_{\mu\nu}$ from the numerically evolved metric before assembling the Ricci and Einstein tensors. The same symbols drive the ray-tracing behind the Event Horizon Telescope's images of M87 and Sgr A: a photon's path through the warped spacetime around a black hole is a null geodesic, integrated through $\ddot x^\lambda+\Gamma^{\lambda}{}_{\mu\nu}\dot x^\mu\dot x^\nu=0$, and the bright ring you see is where those geodesics pile up near the photon sphere. Closer to home, the atomic clocks aboard GPS satellites are corrected for a gravitational redshift that is, at leading order, the $\Gamma^{r}{}_{tt}$ of the weak-field metric integrated over the orbit — get the Christoffel symbol wrong and positions drift by kilometers per day. The frontier open problem is that the metric connection is torsion-free by fiat: theories with torsion (Einstein–Cartan gravity, and the teleparallel reformulations where curvature is traded for torsion entirely) keep the antisymmetric part that the Levi-Civita choice discards, and whether nature uses it is an experimental question still open at the interface of gravity and spin.

Draw a coordinate grid on a curved surface — lines of latitude and longitude on a globe, or circles and rays on a flat sheet of polar graph paper. The basis vectors $\partial_r$ and $\partial_\phi$ literally point in different directions and carry different lengths at different points: $\partial_\phi$ at the rim of a circle is longer than $\partial_\phi$ near the center, because a step of one radian in $\phi$ sweeps a longer arc when $r$ is large. The Christoffel symbols are the precise ledger of that tipping and rescaling. They answer the question, "if I pick up a basis vector here and slide it one infinitesimal step, how much has it turned and grown, so that I can subtract off that spurious change and isolate the real change in a vector field?" This is why nonzero $\Gamma$ does not by itself mean gravity. Flat two-dimensional space in polar coordinates has $\Gamma^{r}{}_{\phi\phi}=-r$ and $\Gamma^{\phi}{}_{r\phi}=1/r$ — perfectly nonzero — yet the Riemann tensor vanishes everywhere. The symbols are recording the curvilinear grid, not any curvature of the space. The first figure makes this concrete: the basis vectors fan and stretch across a flat polar sheet, and the symbols quantify exactly that.

Now think of the equivalence principle. In a freely falling elevator, physics looks special-relativistic and gravity vanishes — no ball drops, no floor pushes up. Mathematically that freely falling observer can always choose coordinates in which $\Gamma^{\lambda}{}_{\mu\nu}=0$ at their location and the metric equals $\eta_{\mu\nu}$ there. This is possible precisely because $\Gamma$ is not a tensor: a genuine tensor that is zero in one frame is zero in all, but the inhomogeneous piece in the $\Gamma$ transformation law can be dialed to cancel the symbols at one chosen point. What cannot be dialed away is the second derivative of the metric — the tidal stretching that pulls two nearby free-fall trajectories together or apart. That irreducible content lives in the derivatives $\partial\Gamma$, which assemble the Riemann tensor. Gravity is not the $\Gamma$; gravity is the part of $\partial\Gamma$ that survives every coordinate choice.

Two routes give the same symbols. One grinds the metric formula with its three derivative terms and an inverse-metric contraction; the other writes the geodesic Lagrangian $g_{\mu\nu}\dot x^\mu\dot x^\nu$, runs the Euler–Lagrange equations, and reads the $\Gamma$ off as the coefficients of the velocity products. The second is far faster by hand and automatically respects the lower-index symmetry, which is why it is the working physicist's method.

The master formula is the solution of the fundamental theorem: metric compatibility plus zero torsion, solved for the connection. Everything downstream — geodesics, transport, curvature — is a use of this one expression, so it is worth committing to memory in the form below, with its telltale pattern of two "plus" derivatives and one "minus".

where: $\Gamma^{\lambda}{}_{\mu\nu}$ — connection coefficient (Christoffel symbol of the second kind); $g_{\mu\nu}$ — metric components; $g^{\lambda\sigma}$ — inverse metric, $g^{\lambda\sigma}g_{\sigma\rho}=\delta^{\lambda}{}_{\rho}$; $\partial_\mu=\partial/\partial x^\mu$ — coordinate partial derivative. Physically: the symbol is a weighted gradient of the metric — it is nonzero exactly when the metric components vary from point to point, whether from genuine curvature or merely from a curvilinear grid. The symbols exist to fix differentiation. On a contravariant vector they add a correction with a plus sign; on a covariant one-form the correction flips sign, so that the derivative of a scalar $\omega_\nu V^\nu$ stays an honest partial derivative and the two corrections cancel.

where: $V^\nu$ — contravariant vector components; $\omega_\nu$ — one-form components; $\nabla_\mu$ — covariant derivative, whose output is a genuine tensor. Physically: $\partial_\mu V^\nu$ measures the change of components including the spurious change of the basis; the $\Gamma$ term subtracts the basis change, leaving the true change of the vector. A geodesic is a curve that parallel-transports its own tangent — the straightest possible worldline. Setting the covariant derivative of the tangent along itself to zero gives the equation every free particle and light ray obeys.

where: $\tau$ — affine parameter (proper time for timelike geodesics); $x^\lambda(\tau)$ — the worldline. Physically: the $\Gamma$ term is the entire "gravitational force" in this picture — it is what bends a straight line in coordinates into a falling trajectory, and it can be transformed away locally because gravity is geometry, not a Newtonian force. The non-tensorial character is not a blemish but the source of the equivalence principle. Under a coordinate change the symbols transform with the usual homogeneous tensor piece plus an inhomogeneous term built from the second derivative of the coordinate map.

where: $x'^\mu$ — new coordinates; the final term carries second derivatives of the map. Physically: because that extra term depends only on the coordinate change and not on any field value, it can be tuned to cancel $\Gamma$ at a chosen point — the mathematics of a freely falling frame. The antisymmetric-in-lower-indices part, however, is the genuine torsion tensor and transforms cleanly.

where: $T^{\lambda}{}_{\mu\nu}$ — torsion tensor, set to zero for the Levi-Civita connection. Physically: the vanishing of torsion is exactly the statement that infinitesimal coordinate parallelograms close, and it is what makes the lower index pair symmetric. Tracing one upper against one lower index collapses the three-term formula to a compact logarithmic derivative of the metric determinant — the single most useful identity for divergences and volume elements.

where: $g=\det(g_{\mu\nu})$ — metric determinant. Physically: this turns the covariant divergence into $\nabla_\mu V^\mu=\tfrac{1}{\sqrt{|g|}}\partial_\mu(\sqrt{|g|}\,V^\mu)$, the form used to write conservation laws and the invariant volume $\sqrt{|g|}\,\mathrm{d}^4x$ in curved spacetime. Finally, differentiating and squaring the symbols builds the object that no coordinate choice can remove — the Riemann curvature tensor, the true seat of gravity.

where: $R^{\rho}{}_{\sigma\mu\nu}$ — Riemann curvature tensor, a genuine tensor although assembled from non-tensorial $\Gamma$. Physically: the $\partial\Gamma$ pieces are the tidal content that survives in a local inertial frame, and the $\Gamma\Gamma$ pieces restore tensor character to the whole.

1 · Christoffel symbols of the two-sphere (confidence builder)

The round two-sphere of radius $a$ has metric $\mathrm{d}s^2=a^2(\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2)$. Find all nonzero Christoffel symbols by the Lagrangian method, and evaluate them at colatitude $\theta=\pi/4$.

Step 1. Write the Lagrangian $L=a^2(\dot\theta^2+\sin^2\theta\,\dot\phi^2)$ and vary $\theta$. The Euler–Lagrange equation gives $\tfrac{\mathrm{d}}{\mathrm{d}\tau}(2a^2\dot\theta)-2a^2\sin\theta\cos\theta\,\dot\phi^2=0$, i.e. $\ddot\theta-\sin\theta\cos\theta\,\dot\phi^2=0$. Matching to $\ddot\theta+\Gamma^{\theta}{}_{\mu\nu}\dot x^\mu\dot x^\nu=0$ reads off $\boldsymbol{\Gamma^{\theta}{}_{\phi\phi}=-\sin\theta\cos\theta}$.

Step 2. Vary $\phi$: $\tfrac{\mathrm{d}}{\mathrm{d}\tau}(2a^2\sin^2\theta\,\dot\phi)=0$ expands to $2a^2\sin^2\theta\,\ddot\phi+4a^2\sin\theta\cos\theta\,\dot\theta\dot\phi=0$, i.e. $\ddot\phi+2\cot\theta\,\dot\theta\dot\phi=0$. The factor of 2 splits between the two orderings $\theta\phi$ and $\phi\theta$, giving $\boldsymbol{\Gamma^{\phi}{}_{\theta\phi}=\Gamma^{\phi}{}_{\phi\theta}=\cot\theta}$. All other symbols vanish.

Step 3. Notice the radius $a$ has dropped out entirely — the symbols are scale-free, a direct sign that they encode angular geometry, not size. Evaluate at $\theta=\pi/4$: $\Gamma^{\theta}{}_{\phi\phi}=-\sin\tfrac{\pi}{4}\cos\tfrac{\pi}{4}=-\tfrac12$ and $\Gamma^{\phi}{}_{\theta\phi}=\cot\tfrac{\pi}{4}=1$. At the 45° parallel, $\Gamma^{\theta}{}_{\phi\phi}=-0.5$ and $\Gamma^{\phi}{}_{\theta\phi}=1$, independent of the sphere's radius.

Step 4. Sanity check against the metric formula (1): $\Gamma^{\theta}{}_{\phi\phi}=-\tfrac12 g^{\theta\theta}\partial_\theta g_{\phi\phi}=-\tfrac12\cdot\tfrac{1}{a^2}\cdot 2a^2\sin\theta\cos\theta=-\sin\theta\cos\theta$, matching Step 1 exactly. These same symbols, differentiated once, give $R^{\theta}{}_{\phi\theta\phi}=\sin^2\theta$ and hence the Gaussian curvature $K=1/a^2$ — the sphere is genuinely curved, unlike the flat polar sheet whose symbols also looked nonzero.

2 · Schwarzschild $\Gamma^{r}{}_{tt}$ and the Newtonian limit (fully numeric)

The Schwarzschild metric in geometric units ($G=c=1$) is $\mathrm{d}s^2=-f\,\mathrm{d}t^2+f^{-1}\mathrm{d}r^2+r^2\mathrm{d}\Omega^2$ with $f=1-2M/r$. Compute $\Gamma^{r}{}_{tt}$, form the radial geodesic equation for a slowly moving particle, and evaluate at the surface of the Earth.

Step 1. With the metric diagonal, use the collapsed formula: $\Gamma^{r}{}_{tt}=-\tfrac12 g^{rr}\partial_r g_{tt}=-\tfrac12 f\,\partial_r(-f)=\tfrac12 f\,\partial_r f$. Since $\partial_r f=2M/r^2$, this gives $\boldsymbol{\Gamma^{r}{}_{tt}=\dfrac{M}{r^2}\Big(1-\dfrac{2M}{r}\Big)}$.

Step 2. The radial geodesic equation for a particle moving slowly ($\dot r,\dot\theta,\dot\phi\ll\dot t$) keeps only the $tt$ term: $\ddot r+\Gamma^{r}{}_{tt}\dot t^2\approx0$. For weak fields $r\gg 2M$ we have $f\approx1$ and $\dot t\approx1$, so $\ddot r\approx-\Gamma^{r}{}_{tt}\approx-M/r^2$. Restoring units ($M\to GM/c^2$, $t\to ct$) turns this into $\ddot r\approx-GM/r^2$ — Newton's law of gravity falls straight out of a Christoffel symbol.

Step 3. Plug in the Earth: $GM_\oplus=3.986\times10^{14}\ \mathrm{m^3/s^2}$ and $r=6.371\times10^6\ \mathrm{m}$ give $GM/r^2=9.82\ \mathrm{m/s^2}$ — the surface gravity, reproduced to three figures. The relativistic correction factor is $2GM/(rc^2)=1.39\times10^{-9}$, utterly negligible, which is why Newtonian gravity is superb at the Earth's surface and the full $\Gamma^{r}{}_{tt}$ is needed only near compact objects.

Step 4. Contrast the regimes: at the Earth's surface the correction is a part in $10^9$; at the Sun's photosphere $2GM_\odot/(R_\odot c^2)\approx4\times10^{-6}$, still tiny but enough to bend starlight measurably; at the Schwarzschild radius $r=2M$ the factor $f\to0$ makes $\Gamma^{r}{}_{tt}$ vanish in these coordinates — a coordinate artifact of the horizon, not a physical fact, exactly the kind of $\Gamma$-based illusion the misconception warning cautions against. Earth: $g=9.82\ \mathrm{m/s^2}$, relativistic correction $1.4\times10^{-9}$.

The Christoffel symbols carry the name of a mathematician who never thought about gravity at all. They emerged from a purely geometric question — when are two quadratic differential forms equivalent under a change of variables — and only decades later became the language in which Einstein wrote the gravitational field.

Texts. Misner, Thorne & Wheeler, Gravitation, §8.5, §10.3, §13.4 · Wald, General Relativity, §3.1 · Carroll, Spacetime and Geometry, §3.1–3.3 · Schutz, A First Course in General Relativity, §5.3–5.6 · Hartle, Gravity, §8.4 and the appendix on the geodesic Lagrangian.

Provenance. The two defining conditions are recalled by name from the prerequisite articles — metric compatibility $\nabla_\lambda g_{\mu\nu}=0$ from aq-gr-metric-compatibility and vanishing torsion $T^{\lambda}{}_{\mu\nu}=0$ from aq-gr-torsion-tensor — and the general notion of a connection from aq-gr-affine-connection and aq-gr-covariant-derivative. All equations here are standard textbook results. A verifier should: (1) reproduce the metric formula (1) against MTW Eq. (8.24), Wald (3.1.30), Carroll (3.27); (2) confirm the flat-space polar values $\Gamma^{r}{}_{\phi\phi}=-r$, $\Gamma^{\phi}{}_{r\phi}=1/r$ from $\mathrm{d}s^2=\mathrm{d}r^2+r^2\mathrm{d}\phi^2$; (3) verify the two-sphere values $\Gamma^{\theta}{}_{\phi\phi}=-\sin\theta\cos\theta$, $\Gamma^{\phi}{}_{\theta\phi}=\cot\theta$ and their $\theta=\pi/4$ evaluations $-0.5$ and $1$; (4) check the Schwarzschild $\Gamma^{r}{}_{tt}=(M/r^2)(1-2M/r)$ and its Earth-surface Newtonian limit $9.82\ \mathrm{m/s^2}$ with correction $1.4\times10^{-9}$; (5) confirm the contracted identity (6) via Jacobi's formula and the Riemann formula (7) against Carroll (3.112).

Prerequisites. The Affine Connection · Covariant Derivative · Metric Compatibility · Torsion Tensor. Leads to. The Levi-Civita Connection · The Geodesic Equation · Geodesics as Extremal Paths · Geodesic Deviation Equation · Parallel Transport.

Ask the tutor: Ask me to grind the Christoffel symbols of any metric you like — Schwarzschild, FRW, or a surface of your choosing — and watch the geodesic Lagrangian method beat the brute-force formula step by step.